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Theorem addsrmo 6631
Description: There is at most one result from adding signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)
Assertion
Ref Expression
addsrmo ((A ((P × P) / ~R ) B ((P × P) / ~R )) → ∃*zwvu𝑡((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ))
Distinct variable groups:   𝑡,A,u,v,w,z   𝑡,B,u,v,w,z

Proof of Theorem addsrmo
Dummy variables f g 𝑞 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 enrer 6623 . . . . . . . . . . . . . . . 16 ~R Er (P × P)
21a1i 9 . . . . . . . . . . . . . . 15 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → ~R Er (P × P))
3 prsrlem1 6630 . . . . . . . . . . . . . . . 16 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → ((((w P v P) (𝑠 P f P)) ((u P 𝑡 P) (g P P))) ((w +P f) = (v +P 𝑠) (u +P ) = (𝑡 +P g))))
4 addcmpblnr 6627 . . . . . . . . . . . . . . . . 17 ((((w P v P) (𝑠 P f P)) ((u P 𝑡 P) (g P P))) → (((w +P f) = (v +P 𝑠) (u +P ) = (𝑡 +P g)) → ⟨(w +P u), (v +P 𝑡)⟩ ~R ⟨(𝑠 +P g), (f +P )⟩))
54imp 115 . . . . . . . . . . . . . . . 16 (((((w P v P) (𝑠 P f P)) ((u P 𝑡 P) (g P P))) ((w +P f) = (v +P 𝑠) (u +P ) = (𝑡 +P g))) → ⟨(w +P u), (v +P 𝑡)⟩ ~R ⟨(𝑠 +P g), (f +P )⟩)
63, 5syl 14 . . . . . . . . . . . . . . 15 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → ⟨(w +P u), (v +P 𝑡)⟩ ~R ⟨(𝑠 +P g), (f +P )⟩)
72, 6erthi 6088 . . . . . . . . . . . . . 14 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → [⟨(w +P u), (v +P 𝑡)⟩] ~R = [⟨(𝑠 +P g), (f +P )⟩] ~R )
87adantrlr 454 . . . . . . . . . . . . 13 (((A ((P × P) / ~R ) B ((P × P) / ~R )) (((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → [⟨(w +P u), (v +P 𝑡)⟩] ~R = [⟨(𝑠 +P g), (f +P )⟩] ~R )
98adantrrr 456 . . . . . . . . . . . 12 (((A ((P × P) / ~R ) B ((P × P) / ~R )) (((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ) ((A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ) 𝑞 = [⟨(𝑠 +P g), (f +P )⟩] ~R ))) → [⟨(w +P u), (v +P 𝑡)⟩] ~R = [⟨(𝑠 +P g), (f +P )⟩] ~R )
10 simprlr 490 . . . . . . . . . . . 12 (((A ((P × P) / ~R ) B ((P × P) / ~R )) (((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ) ((A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ) 𝑞 = [⟨(𝑠 +P g), (f +P )⟩] ~R ))) → z = [⟨(w +P u), (v +P 𝑡)⟩] ~R )
11 simprrr 492 . . . . . . . . . . . 12 (((A ((P × P) / ~R ) B ((P × P) / ~R )) (((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ) ((A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ) 𝑞 = [⟨(𝑠 +P g), (f +P )⟩] ~R ))) → 𝑞 = [⟨(𝑠 +P g), (f +P )⟩] ~R )
129, 10, 113eqtr4d 2079 . . . . . . . . . . 11 (((A ((P × P) / ~R ) B ((P × P) / ~R )) (((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ) ((A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ) 𝑞 = [⟨(𝑠 +P g), (f +P )⟩] ~R ))) → z = 𝑞)
1312expr 357 . . . . . . . . . 10 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R )) → (((A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ) 𝑞 = [⟨(𝑠 +P g), (f +P )⟩] ~R ) → z = 𝑞))
1413exlimdvv 1774 . . . . . . . . 9 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R )) → (g((A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ) 𝑞 = [⟨(𝑠 +P g), (f +P )⟩] ~R ) → z = 𝑞))
1514exlimdvv 1774 . . . . . . . 8 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R )) → (𝑠fg((A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ) 𝑞 = [⟨(𝑠 +P g), (f +P )⟩] ~R ) → z = 𝑞))
1615ex 108 . . . . . . 7 ((A ((P × P) / ~R ) B ((P × P) / ~R )) → (((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ) → (𝑠fg((A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ) 𝑞 = [⟨(𝑠 +P g), (f +P )⟩] ~R ) → z = 𝑞)))
1716exlimdvv 1774 . . . . . 6 ((A ((P × P) / ~R ) B ((P × P) / ~R )) → (u𝑡((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ) → (𝑠fg((A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ) 𝑞 = [⟨(𝑠 +P g), (f +P )⟩] ~R ) → z = 𝑞)))
1817exlimdvv 1774 . . . . 5 ((A ((P × P) / ~R ) B ((P × P) / ~R )) → (wvu𝑡((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ) → (𝑠fg((A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ) 𝑞 = [⟨(𝑠 +P g), (f +P )⟩] ~R ) → z = 𝑞)))
1918impd 242 . . . 4 ((A ((P × P) / ~R ) B ((P × P) / ~R )) → ((wvu𝑡((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ) 𝑠fg((A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ) 𝑞 = [⟨(𝑠 +P g), (f +P )⟩] ~R )) → z = 𝑞))
2019alrimivv 1752 . . 3 ((A ((P × P) / ~R ) B ((P × P) / ~R )) → z𝑞((wvu𝑡((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ) 𝑠fg((A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ) 𝑞 = [⟨(𝑠 +P g), (f +P )⟩] ~R )) → z = 𝑞))
21 opeq12 3542 . . . . . . . . . . 11 ((w = 𝑠 v = f) → ⟨w, v⟩ = ⟨𝑠, f⟩)
2221eceq1d 6078 . . . . . . . . . 10 ((w = 𝑠 v = f) → [⟨w, v⟩] ~R = [⟨𝑠, f⟩] ~R )
2322eqeq2d 2048 . . . . . . . . 9 ((w = 𝑠 v = f) → (A = [⟨w, v⟩] ~RA = [⟨𝑠, f⟩] ~R ))
2423anbi1d 438 . . . . . . . 8 ((w = 𝑠 v = f) → ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) ↔ (A = [⟨𝑠, f⟩] ~R B = [⟨u, 𝑡⟩] ~R )))
25 simpl 102 . . . . . . . . . . . 12 ((w = 𝑠 v = f) → w = 𝑠)
2625oveq1d 5470 . . . . . . . . . . 11 ((w = 𝑠 v = f) → (w +P u) = (𝑠 +P u))
27 simpr 103 . . . . . . . . . . . 12 ((w = 𝑠 v = f) → v = f)
2827oveq1d 5470 . . . . . . . . . . 11 ((w = 𝑠 v = f) → (v +P 𝑡) = (f +P 𝑡))
2926, 28opeq12d 3548 . . . . . . . . . 10 ((w = 𝑠 v = f) → ⟨(w +P u), (v +P 𝑡)⟩ = ⟨(𝑠 +P u), (f +P 𝑡)⟩)
3029eceq1d 6078 . . . . . . . . 9 ((w = 𝑠 v = f) → [⟨(w +P u), (v +P 𝑡)⟩] ~R = [⟨(𝑠 +P u), (f +P 𝑡)⟩] ~R )
3130eqeq2d 2048 . . . . . . . 8 ((w = 𝑠 v = f) → (𝑞 = [⟨(w +P u), (v +P 𝑡)⟩] ~R𝑞 = [⟨(𝑠 +P u), (f +P 𝑡)⟩] ~R ))
3224, 31anbi12d 442 . . . . . . 7 ((w = 𝑠 v = f) → (((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) 𝑞 = [⟨(w +P u), (v +P 𝑡)⟩] ~R ) ↔ ((A = [⟨𝑠, f⟩] ~R B = [⟨u, 𝑡⟩] ~R ) 𝑞 = [⟨(𝑠 +P u), (f +P 𝑡)⟩] ~R )))
33 opeq12 3542 . . . . . . . . . . 11 ((u = g 𝑡 = ) → ⟨u, 𝑡⟩ = ⟨g, ⟩)
3433eceq1d 6078 . . . . . . . . . 10 ((u = g 𝑡 = ) → [⟨u, 𝑡⟩] ~R = [⟨g, ⟩] ~R )
3534eqeq2d 2048 . . . . . . . . 9 ((u = g 𝑡 = ) → (B = [⟨u, 𝑡⟩] ~RB = [⟨g, ⟩] ~R ))
3635anbi2d 437 . . . . . . . 8 ((u = g 𝑡 = ) → ((A = [⟨𝑠, f⟩] ~R B = [⟨u, 𝑡⟩] ~R ) ↔ (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R )))
37 simpl 102 . . . . . . . . . . . 12 ((u = g 𝑡 = ) → u = g)
3837oveq2d 5471 . . . . . . . . . . 11 ((u = g 𝑡 = ) → (𝑠 +P u) = (𝑠 +P g))
39 simpr 103 . . . . . . . . . . . 12 ((u = g 𝑡 = ) → 𝑡 = )
4039oveq2d 5471 . . . . . . . . . . 11 ((u = g 𝑡 = ) → (f +P 𝑡) = (f +P ))
4138, 40opeq12d 3548 . . . . . . . . . 10 ((u = g 𝑡 = ) → ⟨(𝑠 +P u), (f +P 𝑡)⟩ = ⟨(𝑠 +P g), (f +P )⟩)
4241eceq1d 6078 . . . . . . . . 9 ((u = g 𝑡 = ) → [⟨(𝑠 +P u), (f +P 𝑡)⟩] ~R = [⟨(𝑠 +P g), (f +P )⟩] ~R )
4342eqeq2d 2048 . . . . . . . 8 ((u = g 𝑡 = ) → (𝑞 = [⟨(𝑠 +P u), (f +P 𝑡)⟩] ~R𝑞 = [⟨(𝑠 +P g), (f +P )⟩] ~R ))
4436, 43anbi12d 442 . . . . . . 7 ((u = g 𝑡 = ) → (((A = [⟨𝑠, f⟩] ~R B = [⟨u, 𝑡⟩] ~R ) 𝑞 = [⟨(𝑠 +P u), (f +P 𝑡)⟩] ~R ) ↔ ((A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ) 𝑞 = [⟨(𝑠 +P g), (f +P )⟩] ~R )))
4532, 44cbvex4v 1802 . . . . . 6 (wvu𝑡((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) 𝑞 = [⟨(w +P u), (v +P 𝑡)⟩] ~R ) ↔ 𝑠fg((A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ) 𝑞 = [⟨(𝑠 +P g), (f +P )⟩] ~R ))
4645anbi2i 430 . . . . 5 ((wvu𝑡((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ) wvu𝑡((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) 𝑞 = [⟨(w +P u), (v +P 𝑡)⟩] ~R )) ↔ (wvu𝑡((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ) 𝑠fg((A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ) 𝑞 = [⟨(𝑠 +P g), (f +P )⟩] ~R )))
4746imbi1i 227 . . . 4 (((wvu𝑡((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ) wvu𝑡((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) 𝑞 = [⟨(w +P u), (v +P 𝑡)⟩] ~R )) → z = 𝑞) ↔ ((wvu𝑡((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ) 𝑠fg((A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ) 𝑞 = [⟨(𝑠 +P g), (f +P )⟩] ~R )) → z = 𝑞))
48472albii 1357 . . 3 (z𝑞((wvu𝑡((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ) wvu𝑡((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) 𝑞 = [⟨(w +P u), (v +P 𝑡)⟩] ~R )) → z = 𝑞) ↔ z𝑞((wvu𝑡((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ) 𝑠fg((A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ) 𝑞 = [⟨(𝑠 +P g), (f +P )⟩] ~R )) → z = 𝑞))
4920, 48sylibr 137 . 2 ((A ((P × P) / ~R ) B ((P × P) / ~R )) → z𝑞((wvu𝑡((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ) wvu𝑡((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) 𝑞 = [⟨(w +P u), (v +P 𝑡)⟩] ~R )) → z = 𝑞))
50 eqeq1 2043 . . . . 5 (z = 𝑞 → (z = [⟨(w +P u), (v +P 𝑡)⟩] ~R𝑞 = [⟨(w +P u), (v +P 𝑡)⟩] ~R ))
5150anbi2d 437 . . . 4 (z = 𝑞 → (((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ) ↔ ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) 𝑞 = [⟨(w +P u), (v +P 𝑡)⟩] ~R )))
52514exbidv 1747 . . 3 (z = 𝑞 → (wvu𝑡((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ) ↔ wvu𝑡((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) 𝑞 = [⟨(w +P u), (v +P 𝑡)⟩] ~R )))
5352mo4 1958 . 2 (∃*zwvu𝑡((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ) ↔ z𝑞((wvu𝑡((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ) wvu𝑡((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) 𝑞 = [⟨(w +P u), (v +P 𝑡)⟩] ~R )) → z = 𝑞))
5449, 53sylibr 137 1 ((A ((P × P) / ~R ) B ((P × P) / ~R )) → ∃*zwvu𝑡((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1240   = wceq 1242  wex 1378   wcel 1390  ∃*wmo 1898  cop 3370   class class class wbr 3755   × cxp 4286  (class class class)co 5455   Er wer 6039  [cec 6040   / cqs 6041  Pcnp 6275   +P cpp 6277   ~R cer 6280
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6406  df-nq0 6407  df-0nq0 6408  df-plq0 6409  df-mq0 6410  df-inp 6448  df-iplp 6450  df-enr 6614
This theorem is referenced by:  addsrpr  6633
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